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A Goal-Oriented Adaptive Moreau-Yosida Algorithm for Control- and State-Constrained Elliptic Control Problems

Part of: Optimality conditions Partial differential equations, boundary value problems Existence theories

Published online by Cambridge University Press:  27 January 2016

Andreas Günther  and
Moulay Hicham Tber
Andreas Günther
Affiliation:
Bereich Optimierung und Approximation, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany
Moulay Hicham Tber*
Affiliation:
Cadi Ayyad University, Av. Abdelkrim Khattabi, B.P. 511–40000–Marrakech, Morocco
*
*Corresponding author. Email:a.guenther@uni-hamburg.de (A. Günther), hicham.tber@gmail.com (M. H. Tber)
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Abstract

In this work, we develop an adaptive algorithm for solving elliptic optimal control problems with simultaneously appearing state and control constraints. The algorithm combines a Moreau-Yosida technique for handling state constraints with a semi-smooth Newton method for solving the optimality systems of the regularized sub-problems. The state and adjoint variables are discretized using continuous piecewise linear finite elements while a variational discretization concept is applied for the control. To perform the adaptive mesh refinements cycle we derive local error estimators which extend the goal-oriented error approach to our setting. The performance of the overall adaptive solver is assessed by numerical examples.

Keywords

Elliptic optimal control problemcontrol and state constraintsMoreau-Yosida regularizationsemi-smooth Newton methodvariational discretizationgoal-oriented adaptivity

MSC classification

Secondary: 49J20: Optimal control problems involving partial differential equations 49K20: Problems involving partial differential equations 65K10: Optimization and variational techniques 65N50: Mesh generation and refinement
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 3 , June 2016 , pp. 426 - 448
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Becker, R. and Rannacher, R., An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer., 10 (2001), pp. 1–102.Google Scholar
[2]Benedix, O. and Vexler, B., A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints, Comput. Optim. Appl., 44 (2009), pp. 3–25.Google Scholar
[3]Bergounioux, M., Haddou, M., Hintermüller, M. and Kunisch, K., A comparison of a Moreau-Yosida based active strategy and interior point methods for constrained optimal control problems, SIAM J. Optim., 11 (2000), pp. 495–521.CrossRefGoogle Scholar
[4]Bergounioux, M. and Kunisch, K., On the structure of the Lagrange multiplier for stateconstrained optimal control problems, Syst. Control Lett., 48 (2002), pp. 169–176.CrossRefGoogle Scholar
[5]Bergounioux, M. and Kunisch, K., Primal-dual strategy for state-constrained optimal control problems, Comput. Optim. Appl., 22 (2002), pp. 193–224.CrossRefGoogle Scholar
[6]Bonnans, J. F. and Shapiro, A., Optimization problems with perturbations: a guided tour, SIAM Rev., 40 (1998), pp. 228–264.Google Scholar
[7]Casas, E., Boundary control of semilinear elliptic equations with pointwise state constraints, SIAM J. Control Optim., 31 (1993), pp. 993–1006.Google Scholar
[8]Casas, E., Control of an elliptic problemwith pointwise state constraints, SIAM J. Control Optim., 24 (1986), pp. 1309–1318.Google Scholar
[9]Chen, X., Nashed, Z. and Qi, L., Smoothing methods and semismooth methods for nondifferentiable operator equations, SIAM J. Numer. Anal., 38 (2000), pp. 1200–1216.Google Scholar
[10]Deckelnick, K. and Hinze, M., A finite element approximation to elliptic control problems in the presence of control and state constraints, Hamburger Beiträge zur Angewandten Mathematik, Universtität Hamburg, Preprint No. HBAM2007-01, 2007.Google Scholar
[11]Dörfler, D., A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), pp. 1106–1124.Google Scholar
[12]Günther, A. and Hinze, M., A posteriori error control of a state constrained elliptic control problem, J. Numer. Math., 16 (2008), pp. 307–322.CrossRefGoogle Scholar
[13]Hintermüller, M. and Hoppe, R. H. W., Goal-oriented adaptivity in control constrained optimal control of partial differential equations, SIAM J. Control Optim., 47 (2008), pp. 1721–1743.Google Scholar
[14]Hintermüller, M. and Hoppe, R. H. W., Goal-oriented adaptivity in pointwise state constrained optimal control of partial differential equations, SIAM J. Control Optim., 48 (2010), pp. 5468–5487.CrossRefGoogle Scholar
[15]Hintermüller, M. and Hoppe, R. H. W., Goal oriented mesh adaptivity for mixed controlstate constrained elliptic optimal control problems, Appl. Numer. Partial Differential Equations, Comput. Methods Appl. Sci., 15 (2010), pp. 97–111.Google Scholar
[16]Hintermüller, M., Ito, K. and Kunisch, K., The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim., 13 (2003), pp. 865–888.Google Scholar
[17]Hintermüller, M. and Kunisch, K., Feasible and noninterior path-following in constrained minimization with low multiplier regularity, SIAM J. Control Optim., 45 (2006), pp. 1198–1221.Google Scholar
[18]Hintermüller, M. and Kunisch, K., PDE-constrained optimization subject to pointwise constraints on the control, the state and its derivative, SIAM J. Optim., 20 (2009), pp. 1133–1156.Google Scholar
[19]Hinze, M., A variational discretization concept in control constrained optimization: the linearquadratic case, Comput. Optim. Appl., 30 (2005), pp. 45–63.Google Scholar
[20]Hoppe, R. H. W. and Kieweg, M., Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems, Comput. Optim. Appl., 46 (2008), pp. 511–533.Google Scholar
[21]Horn, Roger A. and Johnson, Charles R., Matrix Analysis, Cambridge University Press, New York, 1985.Google Scholar
[22]Ito, Kazufumi and Kunisch, Karl, Lagrange Multiplier Approach to Variational Problems and Applications, Advances in Design and Control, SIAM, Philadelphia, 2008.Google Scholar
[23]Meyer, C., Rösch, A. and Tröltzsch, F., Optimal control of PDEs with regularized pointwise state constraints, Comput. Optim. Appl., 33 (2006), pp. 209–228.Google Scholar
[24]Mifflin, R., Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim., 15 (1977), pp. 957–972.Google Scholar
[25]Qi, L. and Sun, And J., A nonsmooth version of Newton's method, Math. Program, 58 (1993), pp. 353–367.Google Scholar
[26]Schiela, A., State constrained optimal control problems with states of low regularity, SIAM J. Control Optim., 48 (2009), pp. 2407–2432.CrossRefGoogle Scholar
[27]Schiela, A. and Günther, A., An interior point algorithm with inexact step computation in function space for state constrained optimal control, Numerische Mathematik, 119 (2011), pp. 373–407.CrossRefGoogle Scholar
[28]Shapiro, A., On uniqueness of Lagrange multipliers in optimization problems subject to cone constraints, SIAM J. Optim., 7 (1997), pp. 508–518.CrossRefGoogle Scholar
[29]Tröltzsch, F., Regular Lagrange multipliers for control problems with mixed pointwise controlstate constraints, SIAM J. Optim., 15 (2005), pp. 616–634.Google Scholar
[30]Tröltzsch, Fredi, Optimal Control of Partial Differential Equations: Theory, Methods and Applications, AMS, 2010.Google Scholar
[31]Vexler, B. and Wollner, W., Adaptive finite elements for elliptic optimization problems with control constraints, SIAM J. Control Optim., 47 (2008), pp. 509–534.Google Scholar
[32]Vogt, W., Adaptive Verfahren zur numerischen Quadratur und Kubatur, IfMath TU Ilmenau, Preprint No. M 1/06, 2006.Google Scholar
[33]Weiser, M., Interior point methods in function space, SIAM J. Control Optim., 44 (2005), pp. 1766–1786.CrossRefGoogle Scholar
[34]Wollner, W., A posteriori error estimates for a finite element discretization of interior point methods for an elliptic optimization problem with state constraints, Comput. Optim. Appl., 47 (2010), pp. 133–159.Google Scholar
[35]Zowe, J. and Kurcyusz, S., Regularity and stability for the mathematical programming problem in Banach spaces, Appl. Math. Optim., 5 (1979), pp. 49–62.Google Scholar